On the convergence rate of the augmented Lagrangian-based parallel splitting method

• Published in: Optimization methods & software Vol. 34; no. 2; pp. 278 - 304
• Main Authors: Wang, Kai, Desai, Jitamitra
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 04-03-2019; Taylor & Francis Ltd
• Subjects: augmented Lagrangian method; Computational geometry; Convergence; convergence rate; Convexity; Decomposition; Ergodic processes; Estimating techniques; global linear convergence; Jacobian decomposition; Mathematical analysis; Mathematical programming; Nonlinear programming; Optimization; parallel splitting method; separable convex programming; Splitting
• ISSN: 1055-6788; 1029-4937
• DOI: 10.1080/10556788.2017.1370711

The augmented Lagrangian method (ALM) is a well-regarded algorithm for solving convex optimization problems with linear constraints. Recently, in He et al. [On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming, SIAM J. Optim. 25(4) (2015), pp. 2274-2312], it has been demonstrated that a straightforward Jacobian decomposition of ALM is not necessarily convergent when the objective function is the sum of
functions without coupled variables. Then, Wang et al. [A note on augmented Lagrangian-based parallel splitting method, Optim. Lett. 9 (2015), pp. 1199-1212] proved the global convergence of the augmented Lagrangian-based parallel splitting method under the assumption that all objective functions are strongly convex. In this paper, we extend these results and derive the worst-case
convergence rate of this method under both ergodic and non-ergodic conditions, where t represents the number of iterations. Furthermore, we show that the convergence rate can be improved from
to
, and finally, we also demonstrate that this method can achieve global linear convergence, when the involved functions satisfy some additional conditions.